# Verify the Rolle’s Theorem for the function in

Given:

(i) f(x) is continuous in

(ii) $f^{\prime}(x)=e^{x}(\cos x-\sin x)$

f(x) is differentiable in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

(iii)

$f\left(\frac{-\pi}{2}\right)=f\left(\frac{\pi}{2}\right)=0$

All three conditions of Rolle's Theoremare satisfied. So, there exist $c\in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $f^{\prime}(c)=0$.

$f'(c) = e^{c}(\cos c-\sin c)=0$

$\tan c = 1$

$c=\frac{\pi}{4} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

Hence proved.

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